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Proving that a vector field is conservative

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1. The problem statement, all variables and given/known data

Screenshot (236).png


2. Relevant equations

$$F = \nabla \phi$$

3. The attempt at a solution

Let's focus on determining why this vector field is conservative. The answer is the following:

Screenshot (238).png

Screenshot (239).png


I get everything till it starts playing with the constant of integration once the straightforward differential equations have been solved.

May you explain how does it conclude that ##\phi (x, y, z)## is a potential for ##F##?

Thanks.
 

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DrClaude

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You want to find ##\phi## by integrating ##F##. You start by integrating with respect to ##x##, which tells you about the function form of ##\phi## with respect to ##x##, up to a constant ##C_1##, which is a constant wrt ##x## but can be a function of ##y## and ##z##. You then find how ##C_1## changes with ##y## by integrating over ##y##, and so on.
 
263
21
You want to find ##\phi## by integrating ##F##. You start by integrating with respect to ##x##, which tells you about the function form of ##\phi## with respect to ##x##, up to a constant ##C_1##, which is a constant wrt ##x## but can be a function of ##y## and ##z##. You then find how ##C_1## changes with ##y## by integrating over ##y##, and so on.
But I do not understand why ##\frac{\partial \phi}{\partial y} = \frac{\partial C_1}{\partial y}##
 

vela

Staff Emeritus
Science Advisor
Homework Helper
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At that point in the solution, what is ##\phi## equal to? What do you get when you differentiate it with respect to ##y##?
 

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